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Which of the following statements is/are true?
a. A perfect square always has an odd numbers of factors.
b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.
c. 72 has 11 factors.
- a)(a) and (b)
- b)(b) and (c)
- c)(a) and (c)
- d)All the statements are true
- e)None of the statements is true
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Which of the following statements is/are true?a. A perfect square alwa...
A perfect square is of the form n2a Number of factors = (2a + 1), which is odd Hence, statement (a) is correct.
Hence, options 2 and 5 can be eliminated.
Since statement (c) is easier to check using a direct formula, check for statement (c) first.
Hence, options 2 and 5 can be eliminated.
Since statement (c) is easier to check using a direct formula, check for statement (c) first.
72 = 23 x 32
Number of factors = (3 + 1)(2 + 1) = 12 Thus, statement (c) is incorrect.
Hence, options 3 and 4 can be eliminated.
Hence, option 1.
Note: Statement (a) can also be checked by considering 2 or 3 random perfect squares and finding their number of factors.
Statement (b) can be verified as under:
Hence, options 3 and 4 can be eliminated.
Hence, option 1.
Note: Statement (a) can also be checked by considering 2 or 3 random perfect squares and finding their number of factors.
Statement (b) can be verified as under:
N is a perfect square and has 5 factors between 1 and √n.
Half of the factors of a number which is a perfect square are lesser than its square root and half of them are greater than its square root. e.g., 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18 and 36 and √36= 6 1,2, 3 and 4 are less than 6 while 9, 12, 18 and 36 are greater than 6. If the number of factors of n between 1 and √n = 5, then the number of factors between √n and n is also 5.
Half of the factors of a number which is a perfect square are lesser than its square root and half of them are greater than its square root. e.g., 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18 and 36 and √36= 6 1,2, 3 and 4 are less than 6 while 9, 12, 18 and 36 are greater than 6. If the number of factors of n between 1 and √n = 5, then the number of factors between √n and n is also 5.
Apart from this, 1, √n and n are also factors of n.
Total number of factors = 5 + 5 + 3=13.
Total number of factors = 5 + 5 + 3=13.
Most Upvoted Answer
Which of the following statements is/are true?a. A perfect square alwa...
Statement (a): A perfect square always has an odd number of factors.
To determine the number of factors a perfect square has, we need to consider its prime factorization. Let's take the example of a perfect square, say 36.
Prime factorization of 36: 2^2 * 3^2
To find the number of factors, we add 1 to the powers of each prime factor and multiply them together.
Number of factors of 36 = (2 + 1) * (2 + 1) = 3 * 3 = 9
In this case, the number of factors is odd (9). However, it is not always the case that a perfect square will have an odd number of factors. For example, consider the perfect square 16.
Prime factorization of 16: 2^4
Number of factors of 16 = (4 + 1) = 5
In this case, the number of factors is odd (5). Hence, statement (a) is true.
Statement (b): If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.
To understand this statement, let's consider an example. Suppose n is a perfect square and it has five factors between 1 and the square root of n. Let's say the square root of n is x, and we have factors a, b, c, d, and e where 1 < a="" />< b="" />< c="" />< d="" />< e="" />< />
The factors of n can be represented as (a, b, c, d, e, f, g, h, i, j, k, l, m), where f, g, h, i, j, k, l, and m are factors greater than x.
Now, we need to find the total number of factors of n.
Number of factors of n = (number of factors between 1 and x) * (number of factors greater than x)
Since there are five factors between 1 and x, the factor (f, g, h, i, j) can be any of the five factors, and the remaining factors (k, l, m) can be any of the factors greater than x.
Hence, the total number of factors of n = 5 * (number of factors greater than x)
Since the number of factors greater than x is not specified, we cannot determine the exact number of factors of n.
Therefore, statement (b) is false.
Statement (c): 72 has 11 factors.
To determine the number of factors of 72, we need to consider its prime factorization.
Prime factorization of 72: 2^3 * 3^2
To find the number of factors, we add 1 to the powers of each prime factor and multiply them together.
Number of factors of 72 = (3 + 1) * (2 + 1) = 4 * 3 = 12
In this case, the number of factors is 12, not 11. Hence, statement (c) is false.
Based on the explanations above, only statement (a) is true.
To determine the number of factors a perfect square has, we need to consider its prime factorization. Let's take the example of a perfect square, say 36.
Prime factorization of 36: 2^2 * 3^2
To find the number of factors, we add 1 to the powers of each prime factor and multiply them together.
Number of factors of 36 = (2 + 1) * (2 + 1) = 3 * 3 = 9
In this case, the number of factors is odd (9). However, it is not always the case that a perfect square will have an odd number of factors. For example, consider the perfect square 16.
Prime factorization of 16: 2^4
Number of factors of 16 = (4 + 1) = 5
In this case, the number of factors is odd (5). Hence, statement (a) is true.
Statement (b): If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.
To understand this statement, let's consider an example. Suppose n is a perfect square and it has five factors between 1 and the square root of n. Let's say the square root of n is x, and we have factors a, b, c, d, and e where 1 < a="" />< b="" />< c="" />< d="" />< e="" />< />
The factors of n can be represented as (a, b, c, d, e, f, g, h, i, j, k, l, m), where f, g, h, i, j, k, l, and m are factors greater than x.
Now, we need to find the total number of factors of n.
Number of factors of n = (number of factors between 1 and x) * (number of factors greater than x)
Since there are five factors between 1 and x, the factor (f, g, h, i, j) can be any of the five factors, and the remaining factors (k, l, m) can be any of the factors greater than x.
Hence, the total number of factors of n = 5 * (number of factors greater than x)
Since the number of factors greater than x is not specified, we cannot determine the exact number of factors of n.
Therefore, statement (b) is false.
Statement (c): 72 has 11 factors.
To determine the number of factors of 72, we need to consider its prime factorization.
Prime factorization of 72: 2^3 * 3^2
To find the number of factors, we add 1 to the powers of each prime factor and multiply them together.
Number of factors of 72 = (3 + 1) * (2 + 1) = 4 * 3 = 12
In this case, the number of factors is 12, not 11. Hence, statement (c) is false.
Based on the explanations above, only statement (a) is true.
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Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?
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Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?.
Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is/are true?a. A perfect square always has an odd numbers of factors.b. If a number n has five factors between 1 and the square root of n (n is a perfect square), then n has 13 factors.c. 72 has 11 factors.a)(a) and (b)b)(b) and (c)c)(a) and (c)d)All the statements are truee)None of the statements is trueCorrect answer is option 'A'. Can you explain this answer?.
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